2012 IEEE/RSJ International Conference on Intelligent Robots and Systems October 7-12, 2012. Vilamoura, Algarve, Portugal

Static analysis of parallel robots with compliant joints for in -hand manipulation

Júlia Borràs and Aaron M. Dollar, Member, IEEE

S a Abstract — Many robotic hands use compliant joints because they provide several advantages when interacting with objects in unknown environments, but they also modify the relation between external and internal forces and vary the reachable workspace. This work proposes a detailed study of how compliant joints modify the statics of hands, from the point of view of parallel manipulators. The chosen mathematical framework clarifies the role of joint compliance and its effect on the manipulator performance. This framework is then used in an example application to quantify the reduction/increase of torque exerted by the active joints due to the influence of the passive compliant ones for a three fingered hand.

I. INTRODUCTION Dexterous in-hand manipulation is an important goal for robotics and prosthetics as it greatly expands the utility of the hand past simple grasping and more effectively utilizes the multiple controllable degrees of freedom present in the majority of hands. In general, within-hand manipulation is Fig. 1. Three fingers with spring joints manipulating an object can typically done with the manipulated object held with the benefit from parallel platforms kinetostatic analysis. fingertips in a precision grasp [1, 2], affording the fingers contact forces when grasping an object. Specially in doing the manipulation the largest number of degrees of unknown environments, flexible passive joints can increase freedom and workspace size while maintaining a stable the grasping space and reduce the damage produced for an grasp on the object (Fig. 1). unexpected collision [13]. In addition, m echanisms built The general hand/object configuration during these types with elastic joints o er several other advantages compared of precision manipulation tasks is, in effect, similar to to conventional mechanisms:ff reduction of wear, clearance common parallel mechanisms, in which the fingers become and backlash, compactness, no need for lubrication, the “legs” and the object the “platform”. This observation simplified assembly, reduction of the cost of fabrication, etc. affords us the ability to utilize the deep literature on the The main objective of the paper is to find a framework to mechanics of parallel mechanisms to analyze within-hand deal with compliant joints to perform a kinetostatic analysis. precision dexterous manipulation, enabling, for instance, the As far as the authors are aware, there is no detailed study of manipulable workspace of a given object/hand to be how the use of compliant joints modifies the relationship examined for the purposes of design optimization. Related between the external forces, the configuration of the system work examining within-hand manipulation in the context of and the torques on the joints for parallel mechanisms. To this closed chains mechanisms includes [3-7]. end, this work proposes a mathematical framework to deal Previous work in kinetostatic analysis of parallel with fingers with compliant 1-DOF joints when they are mechanisms, however, has not yet substantially considered manipulating an object. The presented analytical approach is joint compliance, which is a major feature in many hands. In based on parallel robot frameworks [14-16] dealing with particular, underactuated hands [8-11], which are promising compliant joints [17-19]. The developed equations are then for both robotics and prosthetics applications because their applied to an example application in which the reduction or design reduces weight, cost, and complexity, simplifies the increase of the active joint torques due to the effect of the control, and greatly increases durability, commonly utilize passive springs are quantified for specific given compliant joints to reduce the number of actuators and configurations. While stability analysis and the study of the increase the finger adaptability [12]. Compliant passive stiffness matrix are important aspects in the study of joints can perform large deflections, which reduces the compliant parallel manipulators, they are left for future work on the topic.

This work was supported in part by National Science Foundation grant We begin this paper by discussing how compliance has IIS-0952856. been treated in the parallel robots literature to date, leading J. Borràs and A. M. Dollar are with the Department of Mechanical to the presentation of a mathematical framework that clearly Engineering and Materials Science, Yale University, New Haven, CT USA. defines the role of the passive and the active compliant joints (e-mail: {julia.borrassol; aaron.dollar}@yale.edu).

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(section II). Next, we apply this framework to a 3-URS robot manipulator, a structure equivalent to popular underactuated robotic hands such as the Barrett hand [20, 21] or the JPL hand [21], illustrating how the framework might be used in an example application to analyze the reduction/increase of torque in different configurations due to the action of the springs (section III). Finally, in section IV, we discuss the presented results and directions for future work.

II. PARALLEL PLATFORMS WITH COMPLIANT JOINTS Fig. 2. Passive and active joints with and without compliance. A. Background and related work Passive joints are also called unactuated and are free to move. Active joints are controlled by the actuators that exert a torque T. Compliant Several models for parallel manipulators with compliant passive joints exert the torque given by the spring. Compliant active joints have been developed. Many works studying compliant joints exert an actuator torque plus a spring torque. parallel manipulators fall out of the scope of this work a spring with variable free length. The value of the free because they study the structural rigidity of the mechanism, length is the commanded value of the actuator. Thus, the as any manipulator, even with stiff joints, has a structural torque done by an active joint is modeled as compliance that is important for control purposes [22-25].

This is typically done through the study of the stiffness (1) matrix, which can be thought as the Hessian of the potential where is the angle of the active joint, is the energy of the system [26]. commanded value for the actuator in the jth active joint and The static equilibrium equations of the manipulator are is the stiffness constant of the spring. Note that, (1) can be usually obtained using the principle of virtual work: the rewritten into which is the sum of a virtual work done by the applied forces is zero for all the torque done by the spring (with 0 length) plus the torque virtual displacements . It may appear in different equivalent done by the actuator T= (Fig. 2). ways [15, 17, 18, 27], because the work done by a constant Using these definitions, a passive joint does not produce conservative force ( ) is related to the potential force/torque reactions and thus, it does not appear in the energy through the equation . Then, one can static equations. On the contrary, a compliant passive joint equivalently apply the principle of minimum energy: a body exerts force and thus, is typically treated as an active joint shall deform or displace to a position that minimizes the [17, 27, 32], without distinguishing between an actuated total energy . Even more generally, the same conditions can compliant joint and a passive compliant joint. To the best of be obtained using the Lagrangian equations typically used the authors’ knowledge, this distinction has appeared for dynamic models of mechanisms, as done in [18]. Since explicitly in the mathematical formulation only in the in the static case the kinetic energy is zero and non- Quenouelle and Gosselin works [18, 33]. conservative forces are not considered, the Lagrangian equations are equivalent to the minimization of the potential C. Mathematical formulation energy. Consider a parallel manipulator with n degrees of freedom Usually, the static analysis is part of a general numeric and m joints. The vector of all the joint angles is formed procedure and the provided examples are usually planar by manipulators [17, 28-30]. Others study the influence of the (2) external loads on the mechanism configuration [24, 31, 32] where the n active joints fully determine the but there is still confusion between passive/active compliant n degrees of freedom of the platform. The kinematic joints and the influence of the actuators in the configuration constraints are the conditions the joints must hold to of a compliant mechanism. maintain the closed kinematic chains, namely B. The joints of a compliant parallel manipulator (3) and thus, the c= m-n passive joints can be written in terms of Even if all the joints in the fingers of a hand are actuated, the active as once the hand holds an object, close-loop kinematic (4) constraints appear. In the context of parallel manipulators, For that reason, are also called constrained typically some of the joints are left free to move in order to joints. The relation between the active and passive joints will be able to hold these equations true. These are called passive play an important role in the statics using compliant joints joints (also called unactuated). A compliant passive joint is through the matrix G defined as also unactuated, but has a spring and thus, it moves in . (5) accordance to the applied forces (Fig. 2). Active joints are The position and orientation of the platform are actuated and thus, they are assumed to be rigid for a given represented by that can be described with n parameters. configuration. For compliant active joints we use the The forward and inverse kinematic problems consist in definition in [33], that models them as noncompliant obtaining the pose of the platform given the set of actuators connected to a spring, so that they are equivalent to

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generalized coordinates and vice versa, respectively. This external force , subtituting the computed values of into can be done by solving the loop equations: equation (12) leads to a linear system for which can be . (6) computed and finally, the input joint angles that Differentiating the above equation, the Jacobian matrix of the actuators should command can be obtained throug h a parallel manipulator is obtained, relating the velocity of a equation (1). point at the platform with the velocities of the active joints, There are two basic static problems, the forward static (7) analysis, which finds the resultant force generated on the For a parallel platform without compliant joints, the platform in a given configuration, and the inverse static principle of virtual work states that the total work done by a analysis, which finds the equilibrium configurations given wrench applied at the platform must be equal to the total an external force. For a manipulator with compliant joints, work done by the reaction forces on the active joints, that is solving the forward static analysis means solving the linear system in (12). For the inverse static problem, given an , (8) 0 external force F and the motor inputs , the loop equations where is the vector of torques exerted by the active joints. The term can be substituted using the Jacobian in and the static equations must be solved simultaneously to equation (7) leading to find the equilibrium configurations . Observe that the term in equation (12) is the . (9) 0 same as the one obtained without compliance in equation Note that this equation expresses the output force on the (9). Thus, the difference in the static equations when dealing platform in terms of the actuated joint torques, which means with passive compliant joints is the extra term , let us that the platform can resist an external force –F. Usually, in called it . is a wrench acting on the platform resultant the context of parallel robots, the matrix is directly ∗ ∗ computed using screw theory [16, 34, 35]. Although screw from the torques done by the springs in the passive theory may seem an ad hoc methodology, lately there have compliant joints, so, it only depends on the configuration of been several works that show how to use it more the manipulator. systematically [36, 37] . If we want to compute the actuation torques the motors W Consider now the same manipulator with springs in its have to exert to compensate an external wrench , the F passive joints. Such passive compliant joints will appear in output force in terms of the actuator torques must be the static equations because they will exert a force on the and the wrench ∗ will be subtracted from . That platform, and so, they will contribute to the virtual work. Let is, for a given configuration, the system in equation (12) is a linear system that can be rewritten as be the vector of torques done by the active (14) ∗ joints and the vector of torques done by the Using Crammer’s rule, the solution of this linear system can passive joints, so that is the vector of all the torques. This be expressed as means that the term in equation (8) is now ∗ (15) (10) det where has beenδ substitutedδ usingδ equation (5). Then, det Where the notation stands for a matrix obtained from again usingδ the Jacobian definition in equation (7), the static substituting the jth column by a vector . Using determinant equilibrium equation in (9) become multilinear properties, , (11) and thus, the static equations with compliant 0 joints are det det ∗ (16) , (12) det det where for a passive compliant joint i, the magnitude of the ∗ where the first term is the torque that would be obtained torques is given by the Hooke’s law without passive compliant joints. The above equation means (13) that the torque done by the active joint , , will be where is the stiffness constant of the spring, is the modified by the compliant joints by the extra term ∗, that angle of the passive joint and is the free length of the spring. The active torques can be modeled as compliant only depends on the configuration of the manipulator and the stiffness constant of the springs. As there is a limit on the joints using equation (1). F and for torques that the joints can resist, the term can change the are called the external parameters and and the internal 1 ∗ parameters of the manipulator. limits of the reachable workspace. In conclusion, although both the singularities and the D. Considerations on compliant versus non-compliant position workspace will remain the same after substituting joints the joints by spring joints, the reachable workspace will When using compliant active joints, the forward/inverse change due to equation (16). kinematics must always be solved taking into account the external force applied at the platform. For example, given a III. CASE STUDY : 3-URS PLATFORM desired position , the loop equations in (6) can be solved to Consider the manipulator in Fig. 1. It is a 6-DoF find out the corresponding joint angles . For any given manipulator with 3 equal legs with 3 rotational joints each.

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Fig. 4. Points lie always on the circumference at the top of the double cone and thus they can be described as in equation (19 ). Let the position and orientation of the platform be given

by a position vector and a rotation matrix . Then, the coordinates of the attachments with respect to the fixed reference frame located at the center of the base are and (18) On the other hand, the coordinates of the platform Fig. 3. Description of the joints and the geometry of the attachments can also be parameterized with respect to the manipulator in Fig. 1. angles of rotation of the joint angles following Fig. 4. The Let be the axis of rotation of the first joint of point lies always on a circumference at the top of a leg i, with 001 rotation angle . The axis of rotation of the double cone generated by the revolution of the legs around second joint is with a the axis , as shown in Fig. 4. Following this scheme, the rotation angle . Finally, the third axis is parallel 0 to the coordinates of the platform attachments with respect to the previous one, with angle of rotation (see Fig. 3 and 5). fixed reference frame can be written as (19) For this example, the angles are left as passive and the 001 0 active joints are considered without compliance. Then, where following equation (2), the active joints are (20) and the passives . , The attachments at the base are named and the and and are the lengths of the links of the ith leg. corresponding attachment at the platform . Their local Following the steps in Section II.C, the constraint coordinates with respect to the center of the base and the equations are given by the distance constraints between the center of the platform are platform attachments. Therefore, equation (3) consists of the 3 equations 00 for . (21) , This gives a non-lineal expression 1of K in (3). As it was not 0 possible to compute the analytical expression of C in , equation (4), the matrix in equation (5) was computed 0 (17) , using the chain rule 0 0 (22) 0 where equation (4) was used to substitute in (5). . 0 The loop equations in (6) are the 9 equations obtained by The parameters and define the radius of the base and of equating the platform points computed with respect the the platform.

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∗ , 01 ∗ , 0 01 (25) ∗ , 0 01 ∗ ∗ ∗ , 1 0 where the values are all in Nm . That means, for example, that if we put ~1 kg of weight on the platform (an external 1 pure force is applied), the torques 10 that each actuator 00has to 10 exert 000 to compensate such force are Without compliant joints With compliant joints 0

0 10 0

0 1 10 Fig. 5. Scheme of the manipulator in a given configuration with a 10 0 force applied at the center of the platform and the axes of rotation of each leg i andjoint j labeled as . On the left side, the position and 0 0 orientation of the platform are given by the position vector and the 0 0 rotation matrix, and belowthe corresponding joint angles. So, the overall torque0 done by the active joints, 0 computed as position and orientation of the platform (equation (18)) the sum of the absolute values of each torque, is reduced if with the same points computed with the angle joints the manipulator has passive compliant joints. If we choose a (equation (19)). In other words, different stiffness constant, k, then the torques for the for (23) manipulator using compliant joints are 0 1 The Jacobian matrix can be computed using screw theory following the steps proposed in [15]. Depending on and so, we could obtain 10 zero torque 0 on the active joints for a which are the joints chosen as active, different Jacobian constant of . Of course, this is only optimum matrices are obtained. For this example, the angles are 1 left as passive, thus, the Jacobian matrix is for this configuration and this external force, but optimizing equation (16) over the workspace for a range of forces can (24) help to decide the optimal stiffness constant for the passive . joints. where the screws are defined as The set of all pure forces of magnitude 10N that can be applied at the center of the platform are plotted in Fig. 6. Each point on the sphere represents the direction of the , and the elements of the diagonal matrix are force, from the point of the sphere to the center of the and platform. For each force, if the compliant joints suppose a The singularities of the 3-URS manipulator, those positions for which , are equivalent to the octahedral Stewart-Goughdet platform 0 and have an easy geometric interpretation (check [38, 39] for more details).

A. The role of the passive compliant joints All the equations have been analytically derived using Wolfram Mathematica 8. For the numerical simulations, we chose dimensions and , for . All the lengths 1 are 0 in meters( 1m) and the 0angles in radian( 1rad ). Consider the configuration plotted in Fig. 5. To compute the torques the motors need to exert in each active joint to compensate a given external wrench W, we solve the linear system equation (12) with . If the passive joints are not compliant, we solve (9) instead. The stiffness constants for the passive joint springs are set to and the free lengths to Fig. 6. Each point on the sphere represents a force applied from the For the given configuration, the values of the extra term point on the sphere to the center of the platform. The range of colors torques computed using equation (16) are from the lightest (yellow) to the darkest (blue) represent percentages of decrease/increase of the overall torque, from 10% of decrease (lightest/yellow) to 10% of increase (darkest/blue).

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, , , 0 0 0 0 00 1 0 0 0 1 0

1 10 0 1 1 0 0 1 011

1 10 0 0 1 0 10 1 10

1 10 1 0 01 1100 10 00

Fig. 7. Representation of all the forces of magnitude 10N applied to the center of the platform for different configurations. The orientation of the platform is given by the rotations around of the x, y and z axis ( , , and , and the position vector . Lightest areas (yellow) represent a percentage of decrease of 10% or more of the overall torque, versus darkest areas (blue) that represent an increase of 10% or more. reduction of the overall torque, the corresponding point is reachable workspace and manipulability indexes taking into plotted in light color (yellow), on the contrary, in dark color account the presence of active and passive compliant joints. (blue). The colors change from light to dark depending on Using the same framework, the study of underactuation the percentage of decrease/increase, being the lightest and the stability of the system are future goals that can be (yellow) if the decrease is equal or bigger than 10%, and the addressed through the study of the matrices J and G and darkest if the increase is 10% or more. through the differentiation of equation (12), respectively. Fig. 7 shows a summary of similar spheres for different The quantification of the variation of the torques due to configurations of the manipulator. It is shown how the the presence of passive joints has been expressed in an compliant passive joints can help the performance of the analytical equation that can be used in the future to study the manipulator in some directions but not in others. The left modification of the reachable workspace of the manipulator. figure in Fig. 7 represents the same configuration as in Fig. The presented preliminary numerical results show 6, but with a lower platform height. It shows that, when the coherent behavior and a promising starting point towards the angles are bigger than , the colors are flipped with full comprehension of the role of compliant joints in robotic respect to Fig. 6, but with the same symmetry. Fig 7 central hands. and right show similar computations at different configurations. Away from the center, the symmetry shown REFERENCES before is broken and the ball is split in half light and half [1] M. R. Cutkosky, "On grasp choice, grasp models, and the design of dark. This shows that the springs cannot help for all the hands for manufacturing tasks," Robotics and Automation, IEEE directions of forces, but the spring parameters can be Transactions on, vol. 5, pp. 269-279, 1989. optimized to help for a specific set of directions. [2] I. M. Bullock and A. M. Dollar, "Classifying human manipulation A more detailed study of the variation of the extra terms behavior," in IEEE International Conference on Rehabilitation Robotics , 2011, pp. 532-537. ∗ in equation (16) with respect to the location in the [3] A. Bicchi and D. Prattichizzo, "Manipulability of cooperating robots workspace is needed to evaluate the impact of using with unactuated joints and closed-chain mechanisms," IEEE compliant passive joints in the reachable workspace. Transactions on Robotics and Automation, vol. 16, pp. 336-345, 2000. [4] C. Rosales, J. M. Porta, R. Suarez, and L. Ros, "Finding all valid hand configurations for a given precision grasp," in IEEE International IV. CONCLUSIONS AND FUTURE WORK Conference on Robotics and Automation , 2008, pp. 1634-1640. This work has presented a mathematical framework that [5] L. U. Odhner and A. M. Dollar, "Dexterous manipulation with underactuated elastic hands," in Proceedings of the 2011 IEEE clearly distinguishes between active and passive compliant International Conference on Robotics and Automation, 2011, pp. joints. The chosen mathematical formulation shows a 5254-5260. promising convenient framework for the future analysis of [6] D. Prattichizzo, M. Malvezzi, M. Gabiccini, and A. Bicchi, "On the manipulability ellipsoids of underactuated robotic hands with compliance," Robotics and Autonomous Systems, 2011.

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[7] J. T. Y. Wen and L. S. Wilfinger, "Kinematic manipulability of [32] A. Pashkevich, A. Klimchik, and D. Chablat, "Enhanced stiffness general constrained rigid multibody systems," IEEE Transactions on modeling of manipulators with passive joints," Mechanism and Robotics and Automation, vol. 15, pp. 558-567, 1999. Machine Theory, vol. 46, pp. 662-679, 2011. [8] S. Hiroset and Y. Umetani, "The Development of Soft Gripper for the [33] C. Quennouelle and C. Gosselin, "A Quasi-Static Model for Planar Versatile Robot Hand," Mechanism and Machine Theory, vol. 13, pp. Compliant Parallel Mechanisms," Journal of Mechanisms and 351-359, 1978. Robotics-Transactions of the Asme, vol. 1, 2009. [9] A. Dollar and R. Howe, "The Highly Adaptive SDM Hand: Design [34] J. P. Merlet, "Singular configurations of parallel manipulators and and Performance Evaluation," International Journal of Robotics Grassmann geometry," The International Journal of Robotics Research, vol. 29, pp. 585-597, 2010. Research, vol. 8, pp. 45-56, 1989. [10] L. Birglen, C. Gosselin, and T. Laliberté, Underactuated robotic [35] D. Kanaan, P. Wenger, S. Caro, and D. Chablat, "Singularity Analysis hands vol. 40: Springer Verlag, 2008. of Lower Mobility Parallel Manipulators Using Grassmann–Cayley [11] Y. Kamikawa and T. Maeno, "Underactuated five-finger prosthetic Algebra," IEEE Transactions on Robotics, vol. 25, pp. 995-1004, hand inspired by grasping force distribution of humans," in IEEE/RSJ 2009. International Conference on Intelligent Robots and Systems , 2008, pp. [36] J. S. Dai, Z. Huang, and H. Lipkin, "Mobility of overconstrained 717-722. parallel mechanisms," Journal of Mechanical Design, vol. 128, p. 220, [12] T. Laliberte, L. Birglen, and C. Gosselin, "Underactuation in robotic 2006. grasping hands," Machine Intelligence & Robotic Control, vol. 4, pp. [37] J. Zhao, B. Li, X. Yang, and H. Yu, "Geometrical method to 1-11, 2002. determine the reciprocal screws and applications to parallel [13] A. Dollar and R. Howe, "Towards grasping in unstructured manipulators," Robotica, vol. 27, pp. 929-940, 2009. environments: grasper compliance and configuration optimization," [38] D. M. Downing, A. E. Samuel, and K. H. Hunt, "Identification of the Advanced Robotics, vol. 19, pp. 523-543, 2005. special configurations of the octahedral manipulator using the pure [14] J. P. Merlet, Parallel Robots , Second Edition. ed.: Springer, 2006. condition," The International Journal of Robotics Research, vol. 21, [15] L. W. Tsai, Robot analysis: the mechanics of serial and parallel pp. 147-159, 2002. manipulators : Wiley-Interscience, 1999. [39] J. Angeles, G. Yang, and I. M. Chen, "Singularity analysis of three- [16] J. K. Davidson, K. H. Hunt, and G. R. Pennock, "Robots and screw legged, six-DOF platform manipulators with URS legs," IEEE/ASME theory: applications of kinematics and statics to robotics," Journal of Transactions on Mechatronics, vol. 8, pp. 469-475, 2003. Mechanical Design, vol. 126, p. 763, 2004. [17] H.-J. Su and J. M. McCarthy, "A Polynomial Homotopy Formulation of the Inverse Static Analysis of Planar Compliant Mechanisms," Journal of Mechanical Design, vol. 128, p. 776, 2006. [18] C. Quennouelle and C. Gosselin, "Kinematostatic modeling of compliant parallel mechanisms. Application to a 3-PRRR Mechanism, the Tripteron," Meccanica, vol. 46, pp. 155-169, 2011. [19] C. Quennouelle and C. Gosselin, "A Quasi-Static Model for Planar Compliant Parallel Mechanisms," Transactions of the ASME Journal of Mechanisms and Robotics, vol. 1, 2009. [20] W. Townsend, "The BarrettHand grasper – programmably flexible part handling and assembly," Industrial Robot: An International Journal, vol. 27, pp. 181-188, 2000. [21] J. K. Salisbury and J. J. Craig, "Articulated Hands: Force Control and Kinematic Issues," The International Journal of Robotics Research, vol. 1, pp. 4-17, 1982. [22] M. R. Cutkosky and I. Kao, "Computing and controlling compliance of a robotic hand," IEEE Transactions on Robotics and Automation, vol. 5, pp. 151-165, 1989. [23] N. Simaan and M. Shoham, "Geometric Interpretation of the Derivatives of Parallel Robots' Jacobian Matrix With Application to Stiffness Control," Journal of Mechanical Design, vol. 125, pp. 33-42, 2003. [24] M. Griffis and J. Duffy, "Kinestatic control - a novel theory for simultaneously regulating force and displacement," Journal of Mechanical Design, vol. 113, pp. 508-515, 1991. [25] T. Patterson and H. Lipkin, "Structure of robot compliance," Journal of Mechanical Design, vol. 115, pp. 576-576, 1993. [26] J. Kövecses and J. Angeles, "The stiffness matrix in elastically articulated rigid-body systems," Multibody System Dynamics, vol. 18, pp. 169-184, 2007. [27] J. Duffy, Statics and kinematics with applications to robotics : Cambridge Univ Pr, 2007. [28] C. a. Mattson, L. L. Howell, and S. P. Magleby, "Development of Commercially Viable Compliant Mechanisms Using the Pseudo- Rigid-Body Model: Case Studies of Parallel Mechanisms," Journal of Intelligent Materials Systems and Structures, vol. 15, pp. 195-202, 2004. [29] B. H. Kang, J. T.-y. Wen, N. G. Dagalakis, and J. J. Gorman, "Analysis and Design of Parallel Mechanisms With Flexure Joints," IEEE Transactions on Robotics, vol. 21, pp. 1179-1185, 2005. [30] H. H. Pham and I. m. Chen, "Kinematics , Workspace and Static Analyses of 2-DOF Flexure Parallel Mechanism," in 7th International Conference on Control, Automation, Robotics and Vision. , 2002, pp. 968-973. [31] D. Zhang and C. Gosselin, "Kinetostatic modeling of parallel mechanisms with a passive constraining leg and revolute actuators," Mechanism and Machine Theory, vol. 37, pp. 599-617, 2002.

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